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Birthday Probabilities

Tool to calculate the birthday paradox problem in probabilities. How many people are necessary to have 50% chance that 2 of them share the same birthday.

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Tag(s) : Statistics, Fun/Miscellaneous

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# Birthday Probabilities

### Probability to share a same birthday

 Probability That at least N people that exactly N people That not any one
 To be born a same day (at any date) a given date (ie. May 1st)

### Format

 Probability Format Numeric (between 0 and 1) Percentage (in %) Fraction A/B (exact or approximate) Odds (About ≈) X in N Chance

### What is the birthday paradox? (Definition)

The paradox of birthdays is a mathematical problem put forward by Von Mises, who looks for the value N in the problem: In a group of N people there is 50% chance that at least 2 people in the group share the same birthday (day + month). The answer is N = 23, which is quite counterintuitive, hence the paradox.

### What are the hypothesis made to calculate the birthday probabilities?

During the calculation of the birthdate paradox, it is supposed that births are equally distributed over the days of a year (it is not exactly true in reality). In the following, a year has 365 days (leap years are ignored).

### What is the probability for a person to be born a given day of the year?

Probability is 1/365 = 0.0027 = 0.27%, indeed, one chance out of 365 to be bord a precise day.

### What is the probability for a person to be born a different day of mine?

The probability is 364/365 = 0.9973 = 99.73%, indeed, he must be born a distinct day of mine, so there is 364 possibilities out of 365.

### What is the probability for a person to be born the same day as me?

This calculation is the same as What is the probability for a person to be born a given day of the year? This given day is my birthday.

Example: 1/365 = 0.0027 = 0.27%

The calculation could also be formulated by noting that the probability for a person to be born the same day as me is the opposite as the probability for a person to be born a different day than mine.

Example: 1 - 364/365 = 1 - 0.9973 = 0,0027 = 0.27%

It is also the probability that a mother has a second child born on the same day (not the same year) as the first.

### What is the probability for a mother and its child to be born the same day?

This calculation is the same as What is the probability for a person (the child) to be born a given day of the year (the mother birthday)?.

Example: 1 chance in 365 or P = 0.0027 or 0.27%

The probability that a mother and her 2 children were born on the same day

Example: 1 chance in 365 * 365 = 133225 or P = 0.0000075 or 0.00075%

The probability that a mother and her N children were born on the same day is 1 in 365 ^ N

$$P(X) = 1/365^N$$

### What are the odds for 2 people in a group of N to be born the same day?

For N = 2, it is the same as probability for a person to be born the same day as me or the opposite as the probability for a person to be born a different day than mine.

Example: P(N=2) = 1 - (364/365) = 0,0027 = 0.27%

For other people, they have to be born also a distinct day than me, but also a distinct day of the other one (then calculate the opposite):

Example: $$P(N=3) = 1 - (364/365) * (363/365) = 0,0082 = 0.82\% \\ P(N=4) = 1 - (364/365) * (363/365) * (362/365) = 0,0164 = 1.64\% \\ P(N=5) = 1 - (364/365) * (363/365) * (362/365) * (361/365) = 0,0271 = 2.71\%$$

### What is the probability that among 23 people, 2 share the same birth day?

About 0.5, 50% (1 chance out of 2)

### How many people are needed in a group to be sure that 2 share the same birthday?

366 are needed (367 if leap years are taken into account). Indeed, every day of the year plus one are needed to be sure that at least a couple of two people share the same birthday.

### How many people are necessary to have the probability > 50% of 2 people to be born the same given day?

For a given date, 253 people are needed to get a probability of 50% that 2 people to be born a same precise day.

### What is the probability of 2 people among N to be born a given day?

Probabilities are multiplied

Example: P(N=2) = 1 - (364/365) = 0,0027 = 0.27%

Example: P(N=3) = 1 - (364/365)^2 = 0,0055 = 0.55%

Example: P(N=n) = 1 - (364/365)^(n-1)

### What is the probability that N people not to be born a given day of the year?

The probability is

Example: P = (364/365)^N

### How many people are born the same day as myself?

With the hypothesis of a world population of 8 billion people, there are 8000000000 / 365.25 or about 22 million people born on a given day / month.

Limiting to USA (350 millions inhabitants), there are 350000000 / 365.25 or nearly 1 millions US people born on a given day / month date.

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